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2013 Asymptotic Behavior and Stability of Solutions to Barotropic Vorticity Equation on a Sphere
Yuri N. Skiba
Commun. Math. Anal. 14(2): 143-162 (2013).

Abstract

Orthogonal projectors on the subspace $\mathbf{H}_{n}$ of homogeneous spherical polynomials of degree $n$ and on the subspace $\mathbb{P}^{N}$ of spherical polynomials of degree $% n\leq N$ are defined for functions on the unit sphere $S$, and their derivatives $\Lambda ^{s}$ of real degree $s$ are introduced by using the multiplier operators. A family of Hilbert spaces $\mathbb{H}^{s}$ of generalized functions having fractional derivatives of real degree $s$ on $S$ is introduced, and some embedding theorems for functions from $\mathbb{H}^{s} $ and Banach spaces $\mathbb{L}^{p}(S)$ and $\mathbb{L} ^{p}(0,T;\mathbb{X})$ on $S$ are given. Non-stationary and stationary problems for barotropic vorticity equation (BVE) describing the vortex dynamics of viscous incompressible fluid on a rotating sphere $S$ are considered. A theorem on the unique weak solvability of nonstationary problem and theorem on the existence of weak solution to stationary problem are given, and a condition guaranteeing the uniqueness of such steady solution is also formulated. The asymptotic behaviour of solutions of nonstationary BVE as $t\rightarrow \infty $ is studied. Particular forms of the external vorticity source have been found which guarantee the existence of such bounded set $\mathbf{B}$ in a phase space $\mathbf{X}$ that eventually attracts all solutions to the BVE. It is shown that the asymptotic behaviour of the BVE solutions depends on both the structure and the smoothness of external vorticity source. Sufficient conditions for the global asymptotic stability of both smooth and weak solutions are also given.

Citation

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Yuri N. Skiba . "Asymptotic Behavior and Stability of Solutions to Barotropic Vorticity Equation on a Sphere." Commun. Math. Anal. 14 (2) 143 - 162, 2013.

Information

Published: 2013
First available in Project Euclid: 20 December 2012

zbMATH: 1263.76020
MathSciNet: MR3011526

Subjects:
Primary: 76D03 , 76D17 , 76E20

Keywords: asymptotic behavior , asymptotic stability , Incompressible viscous fluid on a rotating sphere , unique solvability

Rights: Copyright © 2013 Mathematical Research Publishers

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Vol.14 • No. 2 • 2013
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